### Klein 2-geometry II

**Update**: I've floated this to the top as some new comments have been added.

This is June's continuation of the attempt to categorify the Erlanger Program. What I think would help enormously is a good candidate for a projective 2-group. The Euclidean 2-group and Poincare 2-group present themselves quite straightforwardly as they are semidirect products of a rotational part and an abelian translational part. So should one be looking to decompose projective transformations in a similar way. There are plenty of stabilizer subgroups to think about - stabilizers of: a point, a line, a point on a line, a point off a line, a triangle, etc.

It might also help to get a feel for smallish 2-groups. One would think that just as there is an adjunction between sets and groups which sends groups to their underlying sets, and sets to the free group with elements as generators, there is a 2-adjunction between categories and 2-groups. I.e., is there such a thing as the 2-group freely generated by the category C? Then we could look to impose relations.

Strict 2-groups are also known as crossed modules and as Cat

^{1}-groups. In his doctoral thesis, Urs Schreiber gives a nice introduction to them in section 3.1. So, a strict 2-group is a pair of groups, G and H, with an action a of G on H, and a homomorphism t from H to G, satisfying the conditions:

a(t(h))(h') = hh'hIn the thesis you see why these conditions are natural.^{-1}

t(a(g)(h)) = gt(h)g^{-1}

In his doctoral thesis, Murat Alp uses computer algebra to calculate the smallest such entities. On pages I-2 and 3, he gives some constructions for forming 2-groups out of two ordinary groups, including these four:

1) H is abelian, the image of t is contained in the centre of G, and G acts trivially on H.

2) H is a normal subgroup of G, t is the inclusion, G acts on H by conjugation.

3) t is a surjection whose kernel lies within the centre of H, and g in G acts on H by conjugation with t

^{-1}g.

4) G is a subgroup of Aut(H) which contains the inner automorphisms of H, and t maps h to conjugation by h.

Yet more good exposition on strict 2-groups, and other interesting stuff on Matt Noonan's site.

Toby Bartels has finished his doctoral thesis - Higher Gauge Theory: 2-bundles. Section 2.4 which discusses 2-groups acting on 2-spaces should be useful for the project.

## 39 Comments:

Butting in a bit here, but on the finite 2-group side I recall seeing a long list of fairly small finite 2-groups, given by crossed modules, listed according to some sort of `2-order' (Alp-Wensley, 2000).

D

Butt in all you like David, and anyone else.

The Alp-Wensley paper you mentioned seems to be here. There's a table in the thesis on the same webpage, beginning on page I-77, which I haven't yet tried to understand.

Urs Schreiber and I are wondering about a categorified linear algebra over here.

Currently I am thinking again about some aspects of representations of 2-groups, in order to get a handle on associated 2-vector bundles.

As I had mentioned some time ago (somewhat hastily) over at the coffee table (->), there is a way to obtain from any ordinary rep of a group H a Vect-linear 2-rep of any strict 2-group with H as its group of morphisms.

It's not particularly deep, but at one point slightly non-obvious. In any case, I haven't seen it discussed anywhere in the literature. Did anyone else?

I have now typed the details a little more cleanly in this pdf

Urs, do you find the same problems with 2-Vect or 2-Hilb as representation 2-categories as Crane and Sheppeard point to on p.3 of this.

Hm, good question. I think what I am talking about here is a little different, but maybe I am making some mistake.

Let's see.

Let C be some monoidal category. Inside the bicategory Mod_C of module categories over C we always find the bicategory Bim(C) of bimodules internal to C.

The injection works as follows.

Send every algebra A internal to C to the category {}_A Mod of C-internal left A modules. This category is a module category under tensoring with C from the right, hence indeed an object of Mod_C.

Next, send every internal A-B bimodule to the morphism of right C-module categories {}_B Mod to {}_A Mod obtained by tensoring (over B) from the left with that bimodule.

Similarly, morphisms of bimodules are sent to natural transformations of these functors.

Now, Kapranov-Voevodsky 2-vector spaces are a subcategory of Bim(C), for C=Vect.

Namely, the n-dimensional KV 2-vector space (the category of n-tuples of ordinary vector spaces) is the category of internal left A-modules, where

A = K \oplus K ... \oplus K (n summands),

with K the ground field we are working over.

Similarly, KV 2-linear maps, which are matrices whose entries are vector spaces, are

K^n-K^m

bimodules, and so on.

Now, as we have discussed before, this category KV2Vect is essentially a categorification of the category of modules of the natural numbers. As a result, there are very few invertible morphisms in KV2Vect. Accordingly, one hardly finds interesting representations on this category.

On the other hand, the construction that I mentioned in my previous comment works in the larger bicategory Bim(Vect), which is still just a proper subcategory of Mod_Vect.

In Bim(C) we have many more invertible 1- and 2-morphisms.

For instance, every morphisms of algebras induces a bimodule and the tensor product of these bimodules corresponds to the composition of morphisms of these algebras.

So, that's why I think the construction I mentioned does not run into the same problems as those you allude to.

But if anyone thinks I am wrong, please let me know.

I was visiting my parents last week - one of the few places in the world without any internet access.

Now that I've returned to civilization, I feel a burning urge to carry to completion our project of trying to categorify Klein's approach to geometry! To do this, I'll sadly have to ignore all the fascinating comments on other subjects....

Unfortunately, I haven't made much progress lately. So, I'll warm up my brain with a little plain old uncategorified Klein geometry.

A while back, David pointed out a serious mistake I made. In a moment of temporary insanity, I said "incidence" of figures in a Klein geometry corresponded to their stabilizer subgroups having a nontrivial intersection. I don't know why I said this, because I really did know better. I've been cutting corners at various points in my description of Klein's Erlanger Programme, but this is going too far! It needs to be fixed, which means fixing a few other things too.

Incidence is not, in general, a yes-or-no affair. The right thing to say is that given two types of geometrical figures, they have a set of different possible "types of incidence", which can be classified in a systematic way using double cosets.

Say our Klein geometry is given by some group G. A "type of figure" is a subgroup H. The "figures of this type" are elements of the set G/H. The group G acts on this set in the obvious way - and it acts transitively. All figures of type H "look the same" up to a symmetry in G.

Now consider figures of type H and also of type H'. The set of

pairsof figures, one of each type, isG/H x G/H'

G acts on this space in an obvious way, but not transitively. That's because not all pairs of figures look "the same" - they can be incident in various way.

The set of types of "incidence" is the quotient space

G\[(G/H) x (G/H')]

where now we are modding out by the left action of G. This is called a space of

double cosets.For example, when we're doing projective plane geometry, G = SL(3). If we take H and H' to be the maximal parabolic subgroups corresponding to "points" and "lines" - David described them earlier - the space of double cosets will have just two elements. This is because a point can either lie on a line, or not lie on a line. There are only two "types of incidence" in this case!

Get it?

(Note that what I'm saying here also corrects another oversimplification I sometimes like to make when talking about "The Basic Principle of Galois Theory". It's

not always truethat a figure is determined by its stabilizer subgroup! For example, when G is abelian, all points in G/H have the same stabilizer. But, there can be lots of different figures of type H.Different figures don't have the same stabilizer when we're talking about points and lines in the projective plane... basically because G is "highly nonabelian". But, for the general philosophy of Klein geometry, we need to worry about this issue.)

So in a less flexible geometry, like Euclidean, this space of double cosets - G\[(G/H) x (G/H')] - can be much larger. In the case of points and lines, a double coset contains point-line pairs for which the point is a specific distance from the line.

We've now got to see how this categorifies. (G/H) x (G/H') is the product category (groupoid)with objects ((point, rotation), (line, how far slid/reflected)). So how does G act on this groupoid? Or, how does G realise itself in Aut ((G/H) x (G/H'))?

If my original hunch about incidence was right, i.e., that two weak lines would meet at a weak point, where angle between lines corresponds to rotatedness of weak point, then we'd expect there to be a relationship between G\[(G/H) x (G/H')] and (G/H''), where H and H' are line stabilizers and H'' is a point stabilizer.

David writes:

So in a less flexible geometry, like Euclidean, this space of double cosets - G\[(G/H) x (G/H')] - can be much larger. In the case of points and lines, a double coset contains point-line pairs for which the point is a specific distance from the line.Exactly! You consider pairs consisting of a point and a line. You mod out by the Euclidean group action on these pairs, and you see the only information left is the

distancebetween the point and the line.Another way to say it is that if

G/H

is the set of

figures of type H, andG/H'

is the set of

figures of type H', then a binary relation between a figure of type H and one of type H' is a subset ofG/H x G/H'

A

G-invariant binary relationbetween such figures is a subset ofG\(G/H x G/H')

Every such relation is the union of a bunch of

pointsinG\(G/H x G/H')

So, a point in here an

atomicorminimalG-invariant binary relation between a figure of type H and a figure of type H'. All more complicated such relations can be built up using the logical connective "or", which corresponds to union. These atomic invariant binary relations can also be calledtypes of incidencebetween figures of type H and type H'.In the case of points and lines in Euclidean plane geometry, these atomic invariant binary relations are all of the form "the point p is of distance d from the line L".

So, every Euclidean-invariant statement you can make about the relation between a point and a line can be built from such statements using only the logical connective "or"! (You may need to use it an infinite number of times.)

In the case of projective geometry, the only types of incidence between a point and a line are "the point p lies on the line L" and "the point p does not lie on the line L". This is why projective geometry is simpler.

Anyway, I'd thought about this intriguing hunch of yours, that in categorified Euclidean geometry

two weak lines would meet at a weak point, where angle between lines corresponds to rotatedness of weak point [....]

but I was stymied because I couldn't see how to determine whether this was "true", not knowing the rules of the game.

But now you're reminding me that the general framework of Klein geometry

determinesthe "types of incidence" possible between two figures, as described above.So, if I can categorify the above general abstract nonsense, the general framework of Klein 2-geometry should

determinethe types of incidence between two figures in categorified Euclidean geometry!Or projective geometry, for that matter.

I think this is the sound way to proceed.

So, now I can go to bed.

It's nice to sleep feeling one has made some progress.

It's nice to sleep feeling one has made some progress.Definitely! Let's see if we can ease tonight's sleep too. In my last comment I was implicitly wondering what a categorified double coset construction looks like, and toying with the thought that it might be a 2-category. But I'm not sure.

Let's treat this in general form. So we have a group G, and subgroups H and K. The standard construction is H\G/K, equivalence classes under the equivalence g ~ g' iff g' = hgk, for some h in H, k in K. Now, has equivalence been treated as identity once or twice.

(a) Categorified H\G/K has objects elements of G, and arrows (h,k) going from g to g' with g' = hgk.

(b) We do things in stages. G/K is a groupoid with arrows g--k-->g' = gk. Then H acts on this groupoid somehow. Perhaps,

g ---k--> gk

|..........|

h..........h

|..........|

V..........V

hg ---k-->hgk

with a 2-morphism for luck.

Then, there might be an equivalence between H\(G/K) and (H\G)/K. Hmm, are double groupoids lurking here?

In our example, we have (G/H x G/H') = (G x G)/(H x H'), so double cosets are: G\(G x G)/(H x H'). We were thinking of G acting on point-line pairs, but if the left action went first would this give a different interpretation?

In section 5 of this paper of Elgueta there are interesting things about general linear 2-groups which we may need at some point.

I was poking about to see whether anything had been done about double cosets and double groupoids in the obvious place - Bangor. I couldn't find anything, but I did see the following appraisal of double cosets on p. 574 of this:

Double coset techniques give examples of deep and wide applications of group theoretic methods in chemistry and physics (Ruch and Klein, 1983). For example, there are applications through Polya’s theory of counting, to considerations of deuterons colliding in scattering theory (Pletsch, 2001). Real semisimple symmetric spaces are often characterised by a pair of commuting involutions of a reductive group and many of their properties are studied in this setting—in this case double cosets are of importance for representation theory of p-adic symmetric K-varieties (Helmink and Brion, 2000). Double coset computation can be seen as a way of constructing finite quotients of HNN-extensions of known groups or as a way of constructing groups given by symmetric presentations(Curtis, 1992).

I never knew there was so much to them.

I put that construction mentioned in the comment two above concerning double groupoids to Ronnie Brown. Here is his reply:

Yours is a good question: what to do about 2 subgroups? here are 3 possible answers:

1) use pullback of fibrations. See section 10.7 of my book. (not symmetric!)

2) use induced actions of categories.

139. (with N. Ghani, A.Heyworth, C.D.Wensley), `String-rewriting systems for double coset systems', J. Symb. Comp. 41 (2006)573-590.

Available from my preprint page, and the arXiv, I think.

This does some real computations!

3) Use global actions (but that is for `interactions' of orbits).

149. (with Bak, A., Minian, G., and Porter, T.), `Global actions, groupoid atlases and applications', J. Homotopy and Related Structures, 63 pages (to appear).

available as a Bangor preprint.

This gives a new `algebraic' context for loacl-to-global.

Any exploration of the use of double groupoids is a good thing, from my viewpoint.

Sonia Natale has papers on the arXiv with her husband on double groupoids and Hopf algebras.

Hope that helps. Best Ronnie

just curious: for an outsider like myself (a theoretical computer scientist), what's a good and gentle introduction to category theory ? I am specially interested in a text that can explain the raison d'etre of category theory: what motivates it, where are good examples of the utility of category theory in mathematics etc. In other words, I'm willing to sacrifice completeness for exposition (but I'm not afraid of rigor :) )

Suresh: You could consider Barr and Wells’s Category Theory for Computing Science . Also, the first half of Colin McLarty’s

Elementary Categories, Elementary Toposesis a clear and gentle introduction to categories. You might like Lawvere and Schanuel’sConceptual Mathematics. I’m not sure that there is a book that altogether meets your requirements, though I expect other people will have good suggestions.Suresh, there are plenty of technical expositions out there, especially for computer scientists, e.g. Awodey's lecture notes. But for an informal overview you might try John Baez's 'Tale of n-Categories' starting here. Perhaps you didn't want to climb above the 1-level, but constructions like adjunctions make more sense from above. And John has no rival in the world of mathematics for ability to convey ideas.

What should we expect the double cosets L\E/P to look like, where L is the stabilizer of a line, G the Euclidean group, and P the stabilizer of a point?

As L\(E/P), it is collecting points at the same distance from a line.

As (L\E)/P, it is collecting lines at the same distance from a point.

Either way, decategorified these are [0,infinity), and the relevant line and point are incident if the coset LeP maps to zero, e the identity element of G.

There's a lot to talk about now, but Lisa is leaving for Wuhan at 2 am tomorrow, so just time for a quick comment now:

David wrote:

In my last comment I was implicitly wondering what a categorified double coset construction looks like, and toying with the thought that it might be a 2-category. But I'm not sure.

This reminds me of a mistake I made. If you have a group acting on a set, the weak quotient is a groupoid. I then thought that if you had a group acting on this groupoid, the weak quotient would be a 2-groupoid.

This is actually true, but the 2-groupoid we get this way is

equivalent to a groupoid... so you might as well just treat it as a groupoid!The guy who told me this was a homotopy theorist, so he put it a different way, something like: "you can't get a space with nontrivial pi_2 by taking the homotopy quotient of a homotopy 1-type by a discrete group". This should be easy to see from the long exact sequence of homotopy groups of a fibration - I think I checked it once, but I would need to check it again to feel 100% sure.

Okay, I have some more time... she's packing. David was talking about double cosets:

So we have a group G, and subgroups H and K. The standard construction is H\G/K, equivalence classes under the equivalence g ~ g' iff g' = hgk, for some h in H, k in K. Now, has equivalence been treated as identity once or twice?

So, as you can tell from my previous post, I'm claiming it

might as wellbe once: H\G/K is just a cute way of writing G/(H x K) where the subgroup H acts on the left of G and K acts on the right. So, it makes sense to categorify H\G/K once, getting the weak quotient G//(H x K). But, one doesn't really gain anything by categorifying it twice! One gets a 2-groupoid which is equivalent to a groupoid.Here's something else I should point out. I've been talking about the set of double cosets

G\(G/H x G/H')

but this is really the same as

H\G/H'

To see this, first note that any left action of a group gives a right action via:

xg := xg^{-1}

So, the left/right business is no big deal for group actions.

So, we get a bijection

G/H x G/H' = H\G x G/H'

Then, when we mod out by G, we can mod out on the left on the left side above, and mod out "on the middle" on the right side, getting a bijection:

G\(G/H x G/H') = H\G/H'

I'm not being terribly clear, I suspect, but a little basic group theory suffices to check what I'm saying.

So, the "special case" of double cosets which we've been discussing:

G\(G/H x G/H')

is really the

general case:H\G/H'

And this means that double cosets always have the conceptual interpretation I mentioned earlier: they are

atomic invariant binary relations. In other words, basic statements you can make about a pair of things on which the group G acts, whose truth is invariant under the action of G, with the property that any such invariant statement is an "or" of these basic ones.As usual, a lot of this is stuff that grew out of conversations with James Dolan, and I'm only scratching the surface....

Good night!

G\(G/H x G/H')

is really the general case:

H\G/H'

So that was why I was going on about L\E/P in my last comment. So what to do next? I suppose L\E/L must be a groupoid whose connected components correspond to incidence relations between a pair of lines. Presumably, possible incidence relations in the Euclidean case are angles of intersection (0, pi/2] and then distances [0, infinity) for parallel lines.

At some point we need to talk about cosets and double cosets of 2-groups, or is there more to think about at the 1-level first?

As usual, a lot of this is stuff that grew out of conversations with James Dolan, and I'm only scratching the surface....It's a great shame he can't be induced to write more. From what I've seen he's a very good expositor. A blog, perhaps.

And this means that double cosets always have the conceptual interpretation I mentioned earlier: they are atomic invariant binary relations.Have people thought about the monoidal category of G-invariant binary relations, which is to G-Set as Rel is to Set?

A composition of 'point is on line' and 'line runs through point' would be 'through two points can be drawn a line'.

Once we have completed the analogy:

Set:Rel::G-Set:G-Rel

We need to categorify:

Groupoid:?::G-Groupoid:?

where G is a 2-group.

So what is a categorifed relation? We discussed this last month, and the best I could come up with was that a 2-relation between two groupoids, C and D, was a sub-groupoid of their cartesian product. Alternatively it's a functor from C to the functor category 2^D.

I have a sneaking feeling that this won't make 2-geometry very interesting. The components of the groupoids will be treated as undifferentiated blocks. One way to think of this is to consider a 2-relation between two copies of the fundamental groupoid of a circle. As a subset of the torus, I think it would have to be everything or nothing.

Perhaps, 2 isn't the right groupoid. After all, it shouldn't be whether a pair of objects are related, but how. Maybe Set would be better.

Might our old friend homotopy theory come to our aid in locating a categorified relation?

The path-connectedness relation between a space and itself sends a pair of points to a truth value.

The path-connectedness 2-relation between a space and itself sends a pair of points to the set of classes of path between them.

On another matter, single elliptic geometry might be easier to think about than Euclidean since to each point (pole)there corresponds a line (equator). Incidence relations before weakening (point-point, point-line, line-line) are then really just angles.

John wrote:

And this means that double cosets always have the conceptual interpretation I mentioned earlier: they are atomic invariant binary relations.

David wrote:

Have people thought about the monoidal category of G-invariant binary relations, which is to G-Set as Rel is to Set?

Yes, at least in certain ways. There's been a lot of work on understanding "relations" in contexts more general than the category of sets. One first attempt was Freyd and Scedrov's theory of allegories. Most people think this was subsumed in the theory of regular categories. A bunch of basic things one wants to do with relations work in any regular category - see for example Prop. 5.8.7 in Taylor's book, or Johnstone's

Elephant. Any topos is a regular category. So, the category GSet (of sets with an action of G) is a regular category. So, relations work nicely in here.However, while this is a quick way to establish a vast number of basic facts about relations invariant under symmetries, it doesn't get at everything interesting about the subject.

In particular, the subject of G-invariant binary relations underlies the theory of Hecke operators and Hecke algebras. This is a little appreciated fact, because most people working on Hecke algebras focus on very special groups G - namely, finite reflection groups or SL(2,Z). Jim has been working out the general theory, and someday I'll have to explain a bit of it on This Week's Finds.

It's also worth noting that G-invariant binary relations and Hecke algebras are part of a bigger picture involving G-invariant n-ary relations and "Hecke PROPs". If we have a symmetry group G and figures of type H, H', H''..., the set of atomic G-invariant n-ary relations is

G\(G x G x G x ...)/(H x H' x H'' x ...)

One could call this the space of

n-tuple cosets. Note that the cute trick that worked in the double coset case, where we rewroteG\(G x G)/(H x H')

as

H\G/H',

no longer works quite so beautifully. So, the term

n-tuple cosetsis sort of a joke - I guess I made it up.I know I should be categorifying Klein geometry, but its so much less work talking about stuff I already know...

I will get to work soon and start categorifying!

David wrote:

What should we expect the double cosets L\E/P to look like, where L is the stabilizer of a line, E the Euclidean group, and P the stabilizer of a point?

As L\(E/P), it is collecting points at the same distance from a line.

As (L\E)/P, it is collecting lines at the same distance from a point.

Either way, decategorified these are [0,infinity), and the relevant line and point are incident if the coset LeP maps to zero, e the identity element of E.

Right!

(I fixed a typo where you wrote "G" for the Euclidean group instead of "E".)

Your study of

L\E/L

is also right. It's the set of atomic Euclidean-invariant binary relations between two lines. These relations are all either of the form

"the lines intersect with angle of intersection equal to theta"

for some angle theta, or

"the lines are parallel with distance equal to d"

for some distance d.

In every case where H\G/H' is interesting, it's also interesting to look at the weak quotient H\\G//H'.

However, right now I'm thinking we should grip the nettle firmly. We've been trying to categorify Euclidean or projective geometry without knowing exactly what 2-group serves as the 2-group of symmetries If we understand the categorified geometry well enough, we might be able to figure out its symmetry 2-group...

...but I haven't been able to do this...

... so maybe it'll be easier in some ways - more

mechanical- tostartby picking a 2-group and then working out its categorified Klein geometry.David wrote:

So what is a categorifed relation? We discussed this last month, and the best I could come up with was that a 2-relation between two groupoids, C and D, was a sub-groupoid of their cartesian product.

Oh! I was hoping you'd do better. Given two elements, a plain old binary relation delivers a mere

truth value- a bit of information saying whether the elements are related or not. That's because being related is a merepropertyof a pair of elements.But you, more than almost anyone in the universe, know how to categorify truth values. You know that a truth value is a (-1)-category, while a set is a 0-category. So, you know that a categorified truth value is a

set... so a categorified property is astructure, since there is a set of ways something can be equipped with a structure.So, to categorify relations I think we want to replace the set of truth values:

2 = {T,F}

by the category of sets:

Set.

And luckily, that's what you decided too!

Perhaps, 2 isn't the right groupoid. After all, it shouldn't be whether a pair of objects are related, but how. Maybe Set would be better.Yes! At least, that's the answer I was fishing for when I raised the question.

A binary relation on a pair of sets S and T is a function

f: S x T -> 2

so one might guess a categorified relation on a pair of categories C and D is a functor

f: C x D -> Set

But, if you think about it, maybe it should be a functor

f: C^{op} x D -> Set

since this is something people already know and love - it's called a

profunctor, and people do think of them as categorified relations.Luckily, when C and D are groupoids, the decision whether to work with the opposite category doesn't make much difference, since we have an equivalence

C = C^{op}

Anyway I reallly like the pattern you hint at above, and I'm hoping it'll play an important role in categorified Klein geometry.

Let's summarize, so everyone gets it:

Given a binary relation f on two sets S and T, an element of S is either related to an element of T or not - being related is a

propertyof this pair of elements. So, we getf: S x T -> 2

Given a

categorified binary relationorprofunctorf on a pair of groupoids C and D, an object of C can be related to an object of D in some set of different ways - being related is an extrastructurewe can put on this pair in various ways. So, a categorified binary relation is a functorf: C x D -> Set

So according to what we did in Minneapolis, profunctors have some connection to the modal logic S5n, a modal logic with different accessibility relations.

Here we're after G-invariant profunctors between G-groupoids, where G is a 2-group. Presumably atomic such things assign singleton or empty sets to pairs of objects in the G-groupoids.

Hazarding a guess, you might think that there is a profunctor in the case of P\\E//P which assigns {[d,a]} to a pair of weak points, where d is the distance between them and a is the angle between their internal arrows. Then the d part gets preserved under the action of the 1-morphism part of E on one side, which just twists the internal arrows there, or in other words, strict points only know about distance.

Profunctors are also called distributors and bimodules. Under the latter name they appear in section 3 of Lawvere's amazing paper METRIC SPACES, GENERALIZED LOGIC, AND CLOSED CATEGORIES.

It also includes the comment that translations are automorphisms at finite distance from the identity (p. 3).

It makes you wonder just how powerful 'generalized logic' might be if it is extended to include n-categorical constructions.

... so maybe it'll be easier in some ways - more mechanical - tostartby picking a 2-group and then working out its categorified Klein geometry.OK, what do you have tucked away on the shelves of your 2-group store? Perhaps we're fed up with Euclidean geometry.

"More generally, given any representation α of a Lie group G on a finite-dimensional vector space V , we can form a Lie crossed module and thus a strict Lie 2-group with this data, taking H = V."

We could choose G = PSL(2, R) acting on R^2.

John wrote:

... so maybe it'll be easier in some ways - more mechanical - to start by picking a 2-group and then working out its categorified Klein geometry.

David replied:

OK, what do you have tucked away on the shelves of your 2-group store? Perhaps we're fed up with Euclidean geometry.

I'm not fed up with categorified Euclidean geometry - I just can't guess what 2-group is underlying it, if any. Same with categorified projective geometry. If I understood "incidence" in this geometry, I could try to figure out the 2-group of "incidence preserving transformations". But, the royal road to understanding incidence uses double cosets... which requires that you have the 2-group to start with!

But what's a good 2-group? They all seem either too complicated or too simple...

This is why I've been dilly-dallying for so long.

But enough. Let's start with some 2-group, and if we don't like it, throw it out.

Any vector space gives a 2-group - in fact a "2-vector space" of the sort Alissa Crans and I studied. It's actually something familiar from basic linear algebra.

When you're first learning about vectors, for example vectors in the plane, it's a bit confusing, because

firstthe teacher says that a vector is apointin the plane, or equivalently an arrow from the origin to that point... andthenthey say a vector is anarrowgoing from one point in the plane to another.If you've ever taught linear algebra, you'll know that this issue leads to many confusions - especially when it's not explained clearly.

What's really going on here is that we're treating the plane as a

2-vector space, with a vector space of points (orobjects) and a vector space of arrows between points (ormorphisms).You can compose arrows by sticking one at the end of the other, just like in any category. Namely: given arrows

p --v--> p'

and

p' --w--> p''

we stick them end to end and get an arrow

p --vow--> p''

People usually call this new arrow "v+w", but I'll write composition as "o", because we mustn't confuse it with the vector space structure on points and arrows. This lets us add points in the plane:

p + q

and it also lets us add arrows

p --v--> p'

and

q --w--> q'

to get an arrow

p+q --v+w--> p'+q'

Get it? I wish I could draw pictures here... it's really simple stuff.

So, we have a category where the objects form a

vector space, the morphisms form avector space, and composition is alinear functionfrom the vector space of composable pairs of arrows to the vector space of arrows! This is precisely a2-vector spacein the sense of Alissa and me (not to be confused with a Kapranov-Voevodsky 2-vector space).Just as a vector space is a special sort of group, a 2-vector space is a special sort of 2-group.

So, let's give it a try.

First, though, let's ponder the Klein geometry associated to a plane old vector space, regarded as a group! It's a little weird, which is why I didn't dive into this example ages ago.

For example, consider the real line, R. What are some subgroups H of R? These correspond to

types of figuresin our Klein geometry. Then, what are the quotient spaces R/H like? These are spaces offigures of type H. Then, what are some double coset spaces H\R/H' like? Points in here aretypes of incidence between figures of type H and figures of type H'.The reason this example is weird is that normal subgroups give weird types of

figures in Klein geometry... and in an abelian group

allsubgroups are normal.To see what I mean, let E be the Euclidean group of the plane. It contains the translation group R^2 as a normal subgroup. What are figures of type R^2? We know by definition that they remain unchanged under any translation! So, they are not "localized" like points and lines, which move when you translate them. What they are is "foliations of the plane by parallel oriented lines". They give the plane a "grain", like the grain in some idealized piece of wood. A "grain" changes when you rotate it, but not when you translate it"!

So, despite their name, normal subgroups give "abnormal" figures in Klein geometry... so an abelian group gives a "completely abnormal" Klein geometry.

But, you're a philosopher, so you know how to keep your wits when things get weird.

When we think of figures being preserved by a subgroup, when do we stop? I mean why do we just say that the rotation group about a point fixes the point and not also the foliation of concentric circles about that point? Or the set of rays through the point.

For example, consider the real line, R. What are some subgroups H of R? These correspond to types of figures in our Klein geometry. Then, what are the quotient spaces R/H like? These are spaces of figures of type H. Then, what are some double coset spaces H\R/H' like? Points in here are types of incidence between figures of type H and figures of type H'.Subsets of R: Z and Q are sufficiently different. R/Z has points in correspondence with a circle. Figures are 1d lattices 1 unit apart. Z\R/Z will amount to much the same. The incidence relations pick up the distance between 1d lattices (should that just be [0,1/2]?)

Then R/Q...

Re the last point, I suppose one could say that distances are asymmetric. If there is a blue 1d lattice and an equally spaced red one at unit intervals, then the distance from blue to red is (1 - that from red to blue), distances in [0,1).

So equivalence classes of R/Z are points on a circle, and these points are *really* groupoids, with Z-many objects. So, a weak point here is a point on a circle indexed by an integer, like a point on a helix, and the incidence relation between 2 points is distance along the helix.

Feels like there's a 2-group acing on all this with G = transformations of helix, H = Z. Might this be where the Euclidean analysis was leading that the generalised incidence relation factors through the incidence relation for strong points? Then the 2-group we wanted there had G = E, H = R^2.

David wrote:

When we think of figures being preserved by a subgroup, when do we stop?

Never! A key feature of mathematics is a somewhat crazed persistence in following the rules we make up. It's this somewhat crazed persistence that leads us to discoveries that normal people would never make.

I mean why do we just say that the rotation group about a point fixes the point and not also the foliation of concentric circles about that point? Or the set of rays through the point.

It makes no real difference. All of these are "the same" for the purposes of Klein geometry! The set of points in the plane is

isomorphic, as a set on which the Euclidean group acts, to the set of foliations by concentric circles. So, any incidence relation that can be defined using one can be defined using the other. So, we're free to use whatever description we like - as long as we're studying incidence geometry.This is the radical nature of Klein's approach to incidence geometry: a

geometryisa group G, atype of figureisa subgroup H, and atype of incidencebetween a figure of type H and one of type H'isa point in H\G/H'.This radical idea is a natural spinoff of taking duality seriously in projective geometry - namely, treating lines in projective geometry on an equal footing with points.

Klein realized what we must do to fully understand this idea. Instead of taking points as fundamental, putting structure on the set of points, and then defining "figures" to be structured sets of points, we should start with the symmetry group G and treat all homogeneous spaces of G on an equal footing, as possible "spaces of figures".

So, Klein geometry studies the group G by studying all homogeneous spaces of G.

But every homogeneous space is of the form G/H. So, if I'm doing Euclidean geometry a la Klein, and I consider the space of all copies of the letter "S" printed in 12-point type on the page, all I care about is that this is a homogeneous space for the Euclidean group, corresponding to a certain subgroup: the stabilizer group of the letter S, namely Z/2. If you prefer to use the letter Z, that's fine: same stabilizer group!

It's clear that such radical thinking soon leads to category theory, in which - as some complain - "things lose their thingness" and we focus on the structure of the category of these things. This is probably why some people whine that category theory is "too abstract".

But the point is, nobody ever

forces usto take these radical simplifying steps - it just turns out to be incredibly useful at times to abstract away a lot of detail and see subjects like geometry from a simplified perspective. If you care about the difference between the letter S and the letter Z, that's perfectly fine: you're just not doing Klein geometry anymore... you're doing some other kind of geometry.When I said

Then the 2-group we wanted there had G = E, H = R^2.Perhaps I meant

Then the 2-group we wanted there had G = E, H = SO(2).if I really want to parallel the R/Z case.

I have a sneaking feeling we ought to be making more of what we said during our exchange about gauge theory.

John wrote:

For example, consider the real line, R. What are some subgroups H of R? These correspond to types of figures in our Klein geometry. Then, what are the quotient spaces R/H like? These are spaces of figures of type H. Then, what are some double coset spaces H\R/H' like? Points in here are types of incidence between figures of type H and figures of type H'.

David wrote:

Subsets of R: Z and Q are sufficiently different.

You mean

subgroupsof R, of course.R/Z has points in correspondence with a circle. Figures are 1d lattices 1 unit apart.

Right, good! In fact, every closed subgroup of R is of the form cZ for some positive number c. The corresponding type of figure is a 1d lattice built of points c units apart. The space of such figures is the quotient R/cZ.

Z\R/Z will amount to much the same. The incidence relations pick up the distance between 1d lattices (should that just be [0,1/2]?)

Good puzzle.

If you imagine a lattice of red dots spaced 1 unit apart, and a lattice of blue dots spaced 1 unit apart, can you tell the difference between "red dots 2/3 of a unit to the right of the blue dots" and "blue dots 2/3 of a unit to the right of the red dots"?

Yes, since the symmetry group of our geometry is just the translation group R: reflections are not symmetries here.So, the double coset space Z\R/Z is not [0,1/2], but [0,1] with its ends identified.

And indeed, this is obvious if we remember that

H\G/H = G/H

when G is abelian!This impliesZ\R/Z = R/Z

so indeed, as you said, "Z\R/Z will amount to much the same" as R/Z.

Note how weird this is:

the space of possible types of incidence between two figures of type H is isomorphic to the space of figures of type H!This is one of those freakish consequences of G beingabelian- it would hardly ever happen in an incidence geometry whose symmetry group was nonabelian.David later wrote:

Re the last point, I suppose one could say that distances are asymmetric. If there is a blue 1d lattice and an equally spaced red one at unit intervals, then the distance from blue to red is (1 - that from red to blue), distances in [0,1).

Whoops! I wrote the above stuff before reading this. Could I have subconsciously glanced at it? Is that why I too used the "red dot/blue dot" trick for figuring out what's going on? Weird!

Anyway: we can have even more fun pondering the ways two lattices with different spacings can be incident:

cZ\R/c'Z

When c and c' are incommensurable, this is a non-Hausdorff space!

Similarly, if we use a nonclosed subgroup of R, we get a non-Hausdorff space of figures of that type, like

R/Q

You can think of this as the space of ways of putting a translated copy of the rationals into the real line.

Personally I think we should put off pondering these non-Hausdorff quotient spaces as long as possible, even though "bad quotient spaces" are a basic theme of Connes' work on noncommutative geometry, and an excellent point of contact between noncommutative geometry and categorification! We want to keep things simple....

David wrote:

So equivalence classes of R/Z are points on a circle, and these points are *really* groupoids, with Z-many objects. So, a weak point here is a point on a circle indexed by an integer, like a point on a helix, and the incidence relation between 2 points is distance along the helix.

Let me see if I understand this. R/Z is the space of unit lattices in the real line, but you're still thinking about our previous attempt to "automatically categorify" Klein geometry through weak quotients, so you're forming the weak quotient R//Z, which is a groupoid. You can think of this as having a helix of objects - unit lattices

equipped with a basepointin a real line - and morphisms from any object on this helix to any other object directly above or below it.Then you're trying to figure out the possible types of incidence between two objects of R//Z. I guess the principled way to do this is to look at Z\\R//Z. We'd then get a

groupoidof types of incidence. Is that what you're doing?It's interesting that we're climbing this far up the ladder of n-categories without even replacing the group R by a 2-group, as I'd been planning....

I should think about this!

Images of grains of wood and Connes-like foliations brings to mind differential forms.

We'd then get agroupoidof types of incidence. Is that what you're doing?That's the idea. Do you think just as

the space of possible types of incidence between two figures of type H is isomorphic to the space of figures of type H!the corresponding groupoids should be equivalent. I.e, the incidence groupoid is equivalent to our R//Z.

In view of the abelianness of everything this shouldn't be surprising.

Yes, my guess is that we have an equivalence of groupoids

Z\\R//Z = R//Z

just as we have an isomorphism of sets

Z\R/Z = R/Z

But, this is the sort of thing we should check. Making things equal twice is the same as making them equal once, but is making them isomorphic in two ways the same as making them isomorphic in one way? No - unless the setup decrees these isomorphisms are equal!

More importantly:

I see that your

reallybold idea is a proposal to get 2-groups acting on groupoids of the form G//H. Automatically boosting Klein geometry to Klein 2-geometry... that would be cool.One annoying thing is that these 2-groups seem to be built using both G and H. To get a 2-group whose "Klein 2-geometry" is a "turn-the-crank categorification" of the Klein geometry associated to G, I'd ideally want to build a 2-group using just G. Or maybe using G and

allits subgroups? Anyway, something functorial in G. This could be too much to ask.You say that:

I have a sneaking feeling we ought to be making more of what we said during our exchange about gauge theory.When was that, where was that, and what did we say? I feel like I've spent my whole life starting in college talking about gauge theory....

In the comments to last month's post:

Isn't this all very reminiscent of gauge theory, and principal bundles?You're just saying that because I started talking about "little dials". But it's true - there's an O(2) bundle over the plane lurking in my description of the "groupoid of points" in 2d Euclidean geometry. The total space of this principal bundle is the space of objects in E(2)//O(2), and O(2) acts on this, and the quotient is the plane. I'm not sure how general this is, and I'm feeling too lazy to work it out now...

... oh, but it's just obvious, I can't help working it out: given any Lie groups G and H, the space of objects of G//H is just G, and this is indeed a principal H-bundle over G/H.

So, yeah - you're right!

The quest continues.

Post a Comment

<< Home